optimal protocol fee

doc/optimal_protocol_fee.md

@@ -12,7 +12,7 @@ Let:
 - $v$ = volume (exogenous, constant)
 - $F$ = total LP fee rate (exogenous, fixed)
 - $f$ = LP fee share (fraction of $F$ retained by LPs)
-- $g$ = protocol fee share (fraction of $F$ retained by protocol)
+- $g$ = protocol fee share (fraction of $F$ retained by the protocol)
 - $r$ = short-term market rate (exogenous, includes risk premium)
 - $L$ = total value locked (endogenous)
 
@@ -25,7 +25,8 @@ In equilibrium, LP earnings equal the opportunity cost:
 $$v f = r L$$
 
 Thus:
-$$L = \frac{vf}{r}$$
+
+$$L = \frac{v f}{r}$$
 
 The TVL adjusts until LPs are indifferent between deploying capital here versus the outside market rate $r$.
 
@@ -33,15 +34,15 @@ The TVL adjusts until LPs are indifferent between deploying capital here versus
 
 Protocol profit is:
 
-$$p = fgL = fg \cdot \frac{vf}{r} = \frac{v}{r} f^2 g$$
+$$p = f g L = f g \cdot \frac{v f}{r} = \frac{v}{r} f^2 g$$
 
 Substituting $f = 1 - g$:
 
-$$p(g) = \frac{v}{r} (1-g)^2 g \quad \text{for } g \in [0,1]$$
+$$p(g) = \frac{v}{r} (1 - g)^2 g \quad \text{for } g \in [0,1]$$
 
-Let $h(g) = g(1-g)^2$. Taking the derivative:
+Let $h(g) = g(1 - g)^2$. Taking the derivative:
 
-$$h'(g) = (1-g)^2 - 2g(1-g) = (1-g)(1-3g)$$
+$$h'(g) = (1 - g)^2 - 2 g (1 - g) = (1 - g)(1 - 3 g)$$
 
 ## 4. Solution
 
@@ -50,16 +51,16 @@ Setting $h'(g) = 0$ yields critical points at $g = 1$ and $g = \frac{1}{3}$.
 Evaluating:
 - $h(0) = 0$
 - $h(1) = 0$
-- $h(1/3) = \frac{1}{3} \cdot \left(\frac{2}{3}\right)^2 = \frac{4}{27}$
+- $h\!\left(\frac{1}{3}\right) = \frac{1}{3} \cdot \left(\frac{2}{3}\right)^2 = \frac{4}{27}$
 
 The maximum occurs at **$g^* = \frac{1}{3}$**, giving:
 
 | Quantity              | Value                     |
-|----------|-------|
+|-----------------------|---------------------------|
 | Protocol fee share    | $g^* = \frac{1}{3}$       |
 | LP fee share          | $f^* = \frac{2}{3}$       |
-| Equilibrium TVL | $L^* = \frac{2v}{3r}$ |
-| Max protocol profit | $p^* = \frac{4v}{27r}$ |
+| Equilibrium TVL       | $L^* = \frac{2 v}{3 r}$   |
+| Max protocol profit   | $p^* = \frac{4 v}{27 r}$  |
 
 ## Conclusion
 
@@ -73,17 +74,13 @@ Under the assumptions:
 the protocol’s optimal fee policy is:
 
 - **Protocol share of the fee pool:**  
-  \[
-  \phi_P^* = \frac{1}{3}
-  \]
+  $\phi_P^* = \frac{1}{3}$
 - **LP share of the fee pool:**  
-  \[
-  \phi_L^* = \frac{2}{3}.
-  \]
+  $\phi_L^* = \frac{2}{3}$
 
 Equivalently, in absolute terms:
 
-- **Optimal protocol fee rate:** \(F/3\),
-- **Optimal LP fee rate:** \(2F/3\).
+- **Optimal protocol fee rate:** $F/3$,
+- **Optimal LP fee rate:** $2F/3$.
 
 This fee split balances extracting revenue from trades against maintaining sufficiently attractive LP returns to support a large equilibrium TVL.